New cosmological constraints with extended-Baryon Oscillation Spectroscopic Survey DR14 quasar sample

We update the constraints on the cosmological parameters by adopting the Planck data released in 2015 and baryon acoustic oscillation (BAO) measurements including the new DR14 quasar sample measurement at redshift z=1.52\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z=1.52$$\end{document}, and we conclude that the six-parameter Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}CDM model is preferred. Exploring some extensions to the Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}CDM model, we find that the equation of state of dark energy reads w=-1.036±0.056\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w=-1.036\pm 0.056$$\end{document} in the wCDM model, the effective number of relativistic degrees of freedom in the Universe is Neff=3.09-0.20+0.18\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_\text {eff}=3.09_{-0.20}^{+0.18}$$\end{document} in the Neff+Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_\text {eff}+\Lambda $$\end{document}CDM model and the spatial curvature parameter is Ωk=(1.8±1.9)×10-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _k=(1.8\pm 1.9)\times 10^{-3}$$\end{document} in the Ωk+Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _k+\Lambda $$\end{document}CDM model at 68%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$68\%$$\end{document} confidence level (C.L.), and the 95%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$95\%$$\end{document} C.L. upper bounds on the sum of three active neutrinos masses are ∑mν<0.16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum m_\nu <0.16$$\end{document} eV for the normal hierarchy (NH) and ∑mν<0.19\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum m_\nu <0.19$$\end{document} eV for the inverted hierarchy (IH) with Δχ2≡χNH2-χIH2=-1.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \chi ^2\equiv \chi ^2_\text {NH}-\chi ^2_\text {IH}=-1.25$$\end{document}.


Introduction
The accuracy of cosmological observations has been significantly improved in the past two decades. The six-parameter CDM model is strongly supported by the precise measurements of anisotropies of the cosmic microwave background (CMB) [1,2]. The Type Ia supernova (SNe) [3,4] and baryon acoustic oscillation (BAO) data [5,6] as a geometric complement directly encode the information of the history of the expansion in the late-time Universe. As an important parameter characterizing today's expansion rate, the Hubble constant is directly measured by the Hubble space telescope (HST) [7]. The BAO measurement is the periodic relic of fluctuations of baryonic matter density in the Universe. It is considered as a standard ruler of the Universe and can be used as an independent way to constrain models. In the previous observations, the BAO is traced directly by galaxies at low redshift and measured indirectly by analysis of the Lyman-α (Lyα) forest in quasar spectra at high redshifts. Recently, the extended-Baryon Oscillation Spectroscopic Survey (eBOSS) [8] released their another percent level BAO measurement at z = 1.52 using the auto-correlation of quasars directly, referred to as DR14 quasar sample [9]. It is a new method to achieve BAO features, which makes DR14 the first BAO distance observations in the range of 1 < z < 2.
The higher redshift at which BAO is measured, the more sensitive to the Hubble parameter. Therefore, we can expect an improvement in constraints on the equation of state (EOS) of dark energy (DE) and a preciser description of the expansion history by including DR14. On the other hand, with increasing total active neutrino mass at fixed θ * , the spherically averaged BAO distance D V (z 1) increases accordingly, but D V (z > 1) falls [10]. It implies that DR14 may improve the constraint on the total active neutrino mass. On the contrary, with increasing the effective number of relativistic species N eff at a fixed θ * and a fixed redshift of matterradiation equality z eq , D V (z) decreases for all BAO measurements [10]. Therefore, DR14 can improve the constraint on N eff as well. In addition, the spatial curvature of our Universe can also be constrained better because the geometry of space affects the detection of the BAO measurement directly and the new released BAO measurement DR 14 fills the gap between 1 < z < 2.
In this paper, we update the constraints on the EOS of DE, the active neutrino masses, the dark radiation and the spatial curvature with the Planck data and the BAO measurements including the DR14 quasar sample at z = 1.52. The paper is arranged as follows. In Sect. 2, we explain our methodology and the data we used. In Sect. 3, the results for different models are presented. Finally, a brief summary and discussion are included in Sect. 4.
To show the BAO data we used, we should introduce the BAO model briefly, which is the basic model of the BAO signal. The volume-averaged values are measured, in [6], by where c is the light speed, D A (z) is the proper angular diameter distance [15], given by and H (z) is and

Results
In this section, we will represent our new constraints on the dark energy, the neutrino masses, the dark radiation and the spatial curvature of the Universe separately.

Constraints on dark energy
In this subsection, we constrain the cosmological parameters in the CDM model, the wCDM model and the w 0 w a CDM model [16,17], respectively. Our results are summarized in Table 1. We run CosmoMC [18] in the CDM model as the basic model, where there are six free cosmological param- Here b h 2 is the density of the baryonic matter today, c h 2 is the cold dark matter density today, 100θ MC is 100 times the ratio of the angular diameter distance to the large scale structure sound horizon, τ is the optical depth, n s is the scalar spectrum index, and A s is the amplitude of the power spectrum of primordial curvature perturbations. The EOS of DE is w = −1.036 ± 0.056 in the wCDM model at 68% confidence level (C.L.). The triangular plot of H 0 , w 0 and w a in the w 0 w a CDM model is shown in Fig. 1 and it indicates that the prediction of CDM is within the 68% confidence region in this figure, which seems to be in conflict with the w 0 , w a values in Table 1. Actually, the probabilities are the integrated probabilities, which means the values in Table 1 have been marginalized over all the other parameters except the aimed parameter. Due to the strong correlation between w 0 and w a , we should check if the prediction of CDM is consistent with datasets in the w 0 − w a 2D contour plot.
Marginalizing over the other cosmological parameters, we also plot the evolution of the normalized Hubble parameter H (z) in Fig. 2 where the Hubble parameter is normalized by comparing with those in the best-fit CDM model.  [7] (named R16) which gives H 0 = 73.24 ± 1.74 km s −1 Mpc −1 . Even though such a tension is slightly relaxed in the wCDM model, it is aggravated in the w 0 w a CDM model. In order to significantly relax such a tension, a more dramatic design of the EOS of DE is needed [19]. In addition, a tension still exists around 2σ on the Hubble parameter at z = 2.34 between the predictions of these three DE models constrained by P15+BAO and the measurement by Lyα forest of BOSS DR11 quasars [20], which gives H (z = 2.34) = 222 ± 7 km s −1 Mpc −1 .

Constraints on the total mass of active neutrinos
The neutrino oscillation implies that the active neutrinos have mass splittings    [21]. That is to say, there are two possible mass hierarchies: if m 1 < m 2 < m 3 , it is a normal hierarchy (NH); if m 3 < m 1 < m 2 , it is an inverted hierarchy (IH).
The neutrino mass spectrum is expressed as Here we set the minimum of the three neutrino masses as a free parameter and the sum of the neutrino masses as a derived parameter. Our results are summarized in Table 2.
The likelihood distribution of m ν for the NH and IH are illustrated in Fig. 3.

Constraints on the dark radiation
The total energy density of radiation in the Universe is given by where ρ γ is the CMB photon energy density, N eff denotes the effective number of relativistic degrees of freedom in the Universe. For the three standard model neutrinos, their contribution to N eff is 3.046 due to non-instantaneous decoupling corrections. Then the additional relativistic degree of freedom N eff ≡ N eff − 3.046 implies the existence of some other unknown sources of relativistic degree of freedom. N eff < 0 is considered to result from incompletely thermalized neutrinos or the existence of photons produced after neutrino decoupling, which is less motivated. But there exist many cases with N eff > 0. If a kind of additional massless particles don't interact with others since the epoch of recombination, their energy density evolves exactly like radiation and thus contributes N eff = 1. There are more explanation for 0 < N eff < 1 considering the non-thermal case and the bosonic particles. The thermalized massless boson decoupled during 0.5 MeV< T < 100 MeV contributes N eff 0.57 and N eff 0.39 if they decoupled before T = 100 MeV [32].
In the N eff + CDM model, N eff is taken as a free parameter. The results are summarized in Table 3. Our results give N eff = 3.09 +0. 18 −0.20 at 68% C.L., which is consistent with the fact that there are only three active neutrinos in the Universe. On the other hand, for example in [7], the dark radiation is proposed to relax the tension on the Hubble constant between the global fitting P15+BAO and the direct measurement by HST. Here we illustrate the constraints on H 0 and N eff in Fig. 4. From Fig. 4, we find that the dark radiation cannot really solve this tension.  According to Eq. (2), the spatial geometry affects the distance measurements, and hence the spatial curvature parameter k can be constrained by using BAO data. In the k + CDM model, k is taken as a free parameter. The constraints on the cosmological parameters in the k + CDM model are given in Table 4. We find that the spatial curvature has been tightly constrained, namely k = (1.8 ± 1.9) × 10 −3 at 68% C.L. and k = (1.8 +3.9 −3.8 ) × 10 −3 at 95% C.L. which is nicely consistent with a spatially flat Universe. Adopting P15 only, the constraint on the spatial curvature is k = (−40 +38 −41 ) × 10 −3 at 95% C.L. which is around one oder of magnitude looser compared with our new result. However, our results improves little compared with the Planck + BAO result in the Planck table, k = (0.2 ± 2.1) × 10 −3 at 68% C.L., which implies that the DR14 sample helps little to constrain the curvature. The constraints on and m are illustrated in Fig. 5.

Summary and discussion
In this paper we provide the new constraints on the cosmological parameters in some extensions to the six-parameter CDM model by combining P15 and BAO data including the DR14 quasar sample measurement released recently by eBOSS. We do not find any signals beyond this cosmological model.
We explore the EOS of DE in two extended models, namely wCDM and w 0 w a CDM model, and find w = − 1.036 ± 0.056 at 68% C.L. in the wCDM model, w 0 = − 0.25 ± 0.32, w a = −2.29 +1.10 −0.91 at 68% C.L. in the w 0 w a CDM model and w = −1 is located within the 68% C.L. region. But the tension on the Hubble constant with the direct measurement by HST and the global fitting P15+BAO in wCDM model cannot be significantly relaxed and the w 0 w a CDM model makes even worse. The neutrino mass normal hierarchy is slightly preferred by χ 2 ≡ χ 2 NH − χ 2 IH = −1.25 compared to the inverted hierarchy, and the 95% C.L. upper bounds on the sum of three active neutrinos masses are m ν < 0.16 eV for the normal hierarchy and m ν < 0.19 eV for the inverted hierarchy. The three active neutrinos are nicely consistent with the constraint on the effective relativistic degrees of freedom with N eff = 3.09 +0. 18 −0.20 at 68% C.L., and a spatially flat Universe is preferred.